This article is cited in 1 paper

RESEARCH ARTICLE

On the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs

Phaisatcha Inpoonjai^a, Thiradet Jiarasuksakun<sup>b

a</sup> Faculty of Sciences and Agricultural Technology, Rajamangala University of Technology Lanna Chiangrai, 99, Sai Khao, Phan District, Chiang Rai, 57120, Thailand
^b Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand

Abstract: Magic rectangles are a classical generalization of the well-known magic squares, and they are related to graphs. A graph $G$ is called degree-magic if there exists a labelling of the edges by integers $1,2,\dots,|E(G)|$ such that the sum of the labels of the edges incident with any vertex $v$ is equal to $(1+|E(G)|)\deg(v)/2$. Degree-magic graphs extend supermagic regular graphs. In this paper, we present a general proof of the necessary and sufficient conditions for the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs. We apply this existence to construct supermagic regular graphs and to identify the sufficient condition for even $n$-tuple magic rectangles to exist.

Keywords: regular graphs, bipartite graphs, tripartite graphs, supermagic graphs, degree-magic graphs, balanced degree-magic graphs, magic rectangles.

MSC: Primary 05C78; Secondary 05B15

Received: 28.12.2016
Revised: 07.03.2017

Language: English